nLab bubble of nothing

Idea

A bubble of nothing in the sense of Witten 82 is an instability of the Kaluza-Klein vacuum where a gravitational instanton mediates the decay of a $5$-dimensional Euclidean Schwarzschild spacetime into nothing: the hole in space grows at the speed of light until it hits null infinity?, leading to the “annihilation” of spacetime.

Construction of the solution

Let us start from the spacetime $(\mathbb{R}^{4}-B_R^3)\times S^1$, where $B_R^3$ is a ball of radius $R$ and $S^1$ is a circle of radius $R_5$, equipped with the following Euclidean Schwarzschild metric:

$$g \;=\; r^2\mathrm{vol}_{S^3} + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2,$$

where $\theta$ is the coordinate on the circle and $r$ is the radius of $\mathbb{R}^{4}$.

Let the volume of the $3$-sphere be $\mathrm{vol}_{S^3} = \mathrm{d}\chi^2 + \sin(\chi)\mathrm{vol}_{S^2}$. We can now go back to the Minkowski signature by the Wick rotation $\psi := -i\chi + i\frac{\pi}{2}$. Thus, we have

$$g \;=\; -r^2\mathrm{d}\psi^2 + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + r^2\cosh^2(\psi)\mathrm{vol}_{S^2} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2.$$

Properties

By change of coordinates $\rho := r\cosh(\psi)$ and $t:=r\sinh(\psi)$, we see that at large radius the metric goes to the Minkowski metric

$$\lim_{r\rightarrow \infty}g \;=\; -\mathrm{d}t^2 +\mathrm{d}\rho^2 + \rho^2\mathrm{vol}_{S^2}+R_5^2\mathrm{d}\theta^2.$$

However, the coordinates $(t,\rho)$ parametrize all the $4$-dimensional base manifold but the region $\{ \rho^2 - t^2 \leq R \}$ of the bubble. The boundary of the bubble in the $4$-dimensional base space has a radius which grows with time as

$$\rho_{\mathrm{BON}}(t) \;=\;\sqrt{R^2+t^2}.$$

Moreover, the effective size of the circle compactification will be depend on $r$ by

$$R_{5}(r) \;=\; R_5\sqrt{1-\frac{R^2}{r^2}},$$

so that it goes to $0$ for $r\rightarrow R$ and it goes to $R_5$ for for $r\rightarrow \infty$.

Related concepts

• Coleman-De Luccia instanton

• Kaluza-Klein theory

• swampland

• instanton

  • gravitational instanton

• double dimensional reduction

• moduli stabilization

• landscape of string theory vacua

  • flux compactification

  • supersymmetry and Calabi-Yau manifolds

  • string theory FAQ – What does it mean to say that string theory has a “landscape” of solutions?

References

• Edward Witten, Instability of the Kaluza-Klein Vacuum, Nucl. Phys. B195 (1982) 481-492 (doi:10.1016/0550-3213(82)90007-4).